This paper investigates the stochastic stability of gyroscopic viscoelastic systems subjected to parametric wide-band noise excitation. The analysis focuses on both moment stability, using moment Lyapunov exponents, and almost-sure stability, via the largest Lyapunov exponent. The wide-band noises considered include Gaussian white noise and Ornstein–Uhlenbeck noise. The Stratonovich stochastic differential equations governing the system with small damping and weak excitation are first converted to Itô stochastic differential equations through stochastic averaging techniques. An elegant mathematical framework is then introduced to approximate the moment Lyapunov exponents through stochastic transformations and an eigenvalue problem. The largest Lyapunov exponent is subsequently derived based on its relationship with the moment Lyapunov exponents. An application example involves deriving the stochastic equations of motion for an axially moving band system with fluctuating tension, analyzing its stochastic stability. The analytical approximations are validated via Monte Carlo simulations and compared with results from the literature. The study also discusses the influence of various parameters on the system’s stability, with potential applications in engineering fields.
This research builds upon Çelik’s earlier study [1], which explored various flow patterns and constructed a bifurcation diagram for the flow with Stokes approximation in a Z-shaped domain driven by dual lids. In this study, enhanced solutions near the re-entrant corner are initially derived using the asymptotic matching technique. These solutions direct attention towards a comprehensive analysis of flow structures and behaviors at the re-entrant corner point. The formation and bifurcation of the separation bubble that appears near the corner is investigated in detail. Subsequently, the (h1, h2) control space diagram, which was previously developed on the basis of cavity height parameters, is recreated for a Reynolds number of 50. Additionally, the impact of the Reynolds number (Re) on vortex formation is analyzed. The findings indicate that at Re = 50, distinct flow patterns occur, and an increase in the Reynolds number accelerates vortex formation, leading to earlier transitions in flow topology and more complex bifurcation scenarios.
From the three-dimensional linear theory of elasticity, two- and one-dimensional descriptions are derived by involving the consistent-approximation approach. The pseudo-reduction technique yields well-known and higher-order theories for quasi two-dimensional and quasi one-dimensional structural members.
In formula (1.1) of the quoted article, the expression of the excenticity eH of ellipses in the Hooke’s problem is wrong. It must be replaced with the following one eH = √2σ/1+σ, where σ = √1−KL2/mE2. The reader is kindly asked to forgive this error which, however, in no way affects the content of the article. Link to the corrected article 10.2298/TAM220213005C
The problem of optimal thrust programming for an intermediate vehicle model is considered. The motion occurs in a vertical plane under a uniform gravitational field, quadratic resistance friction, and thrust force. The control variables are the angle of attack and the thrust force. Phase constraints are imposed on the trajectory inclination angle. It is assumed that the total fuel consumption for thrust control is negligible compared to the vehicle mass, the fuel mass variation does not affect the center of mass dynamics, and a change of the lifting force does not affect the drag force. The region in the space of initial variables for which the problem is solvable is determined, and an optimal synthesis is constructed. It is established that within this domain, the thrust can be maximum, intermediate, or zero. The number and sequence of trajectory arcs with corresponding thrust values and the number of exits to state constrains are determined.
In this paper, we investigate the boundary control of flexible mechanical systems characterized by bending deformation, torsion deformation, and distributed delay. We establish the existence of solutions using the Faedo-Galerkin approach along with energy estimates. Under appropriate assumptions on the delay weight and the proposed control, we demonstrate the exponential stability of the solution via the Lyapunov method.
This paper investigates the wave equation with variable-exponent nonlinearity and logarithmic source term, given by the following: utt − Δu + $ aut|ut|^{m(・)−2} $ = $ bu|u|^{p(・)−2} $ ln |u|, where a and b are positive constants, and the functions m(・) and p(・) satisfy certain required conditions. Using the energy method and several inequality techniques, we establish a finite-time global nonexistence result for specific solutions with positive initial energy, under appropriate conditions. This type of equation has significant applications in various fields, including fluid dynamics, electrorheological fluids, quantum mechanics, nuclear physics, optics, and geophysics.
This paper examines to the global existence of weak solutions for compressible self-gravitating fluids in three-dimensional unbounded domain with a compact Lipschitz boundary, assuming a total finite fluid mass. We prove that there exists a globally defined weak solution that satisfies the energy inequality in differential form.
We investigate a nonsmooth vector fractional continuous-time programming problem with inequality-type phase constraints, motivated by applied-mechanics contexts in which performance is quantified by ratio-type, time-accumulated indices under pointwise-in-time operational constraints, including abrasive machining and grinding, quasi-steady aircraft cruise efficiency, and energy-aware gait scheduling in legged robotics. We derive saddle-point and Karush-Kuhn-Tucker type necessary optimality conditions for properly efficient solutions by combining a continuous-time Slater-type condition with a regularity requirement for convex inequality systems, and we also establish a sufficient saddle-point optimality condition that holds in the convex framework. For models with mixed affine structure, we strengthen the theory beyond classical Slater-based frameworks by introducing two additional verifiable hypotheses, a solvability condition and a separation direction condition. These assumptions yield sharper multiplier conclusions, including nontriviality of multipliers associated with nonaffine constraints, and lead to refined optimality statements without auxiliary parameters. A key lemma is established and provides the main tool underlying these results. We then introduce a vector-valued Lagrangian and formulate a corresponding vector dual model. For the dual problem, we prove weak and strong duality results, including a strong duality theorem that guarantees the absence of a duality gap. Several examples illustrate how the assumptions can be verified and how the theoretical results apply through explicit multiplier constructions.
The investigation of heat transfer within porous medium has attracted considerable interest because of its growing practical importance. The present paper is devoted to the study of a steady-state non-isothermal fluid flow through a vertical cylindrical annulus filled with sparsely packed porous medium. The governing system is given by the Darcy-Brinkman-Boussinesq model where the heat equation includes the viscous dissipation term. The side walls are maintained at uniform temperatures, while the flow is driven by the axial pressure gradient. Introducing the thickness of the annular region as the small parameter of the problem, the goal is to derive the approximation of the flow via asymptotic analysis. Although the governing problem is coupled and nonlinear, the asymptotic solution is proposed in the explicit form and that represents our main contribution. As such, it clearly displays the effects of porous structure and viscous dissipation on the temperature and velocity distribution. Due to the viscous dissipation, we observe the increase in the heat generation leading to a raise in the temperature and velocity profile within the annulus. Moreover, it is deduced that reducing the thickness of the annular region may be employed to strengthen the seepage velocity of the working fluid.
This paper is an outgrowth of the results in the domain of rolling obtained in our recent paper written with F. Silva Leite and I. Markina, and the earlier papers on the rollings of spheres produced with J. Zimmerman. We show that the rolling equations associated with a symmetric semi-Riemannian manifold rolling on its tangent space at a fixed point on the manifold essentially have the same structure as the rolling equations for the n-dimensional sphere rolling on the horizontal hyperplane; that is, we show that the rolling equations are described by a left-invariant distribution D on a Lie group G with the Lie bracket growth D +[D,D] +[D,[D,D]] = TG, reminiscent of the growth (2, 3, 5) for the two spheres rolling on the horizontal plane. We then define rolling geodesics on semi-Riemannian spaces as extensions of sub-Riemannian geodesics in the Riemannian symmetric spaces, and show that the rolling geodesics are the projections of the extremal curves, which, remarkably, are the solution curves of a completely integrable Hamiltonian system in the cotangent bundle of the configuration space. Finally, we illustrate the theory with a few noteworthy examples.
. A mesoscale phase field formulation for high temperature computational mechanics of polycrystalline solids with voids is developed. The mathematical description includes translation and rotation of crystalline grains, diffusion through the crystals and interfaces, lattice growth at the boundary and grain boundary sliding, elastic stresses and compositional eigenstrains. The formulation connects heterologous continua; solids are represented by the lattice continuum, while the voids/gas are the standard mass continuum viscous (and elastically compressible) fluid. The deformation gradient for solids is consequently a state variable with evolution defined in the Eulerian sense. We consider non-inertial, controlled-temperature processes with vacancy-atom exchange diffusion mechanism and isotropic interface energies. Nevertheless, as discussed in the concluding section, the formulation provides the basis for extensions to the processes with significant heat sources or sinks, diffusion of multiple species and anisotropic interface energies.
. Typically, one considers the problem of finding the minimum number of invariants of a dynamical system sufficient for integrability. It can be also assumed that there are invariants not related to integrability that describe other properties of the dynamical system. We compute such tensor invariants for some integrable and non-integrable dynamical systems by using modern computer software and discuss their properties.
. The problem of motion of a material point in a three-dimensional domain bounded by confo cal quadrics is considered. Such a dynamical system is Liouville integrable in the piecewise-smooth sense. For two types of billiard regions, the regions of possible motion of a material point are found, the bifurcation diagrams are constructed, and the semi-local structure of the Liouville foliation is described. On singular Liouville foliation layers of nonzero rank, at least one action variable can be smoothly extended. This consideration, applied to the billiard inside a three-axial ellipsoid, gave a new proof of Staude's construction of an ellipsoid using a thread.
In this paper, we propose a variant of Adler's recutting for projective polygons. We discuss its properties and overview its connection with the cross-ratio relation.
. The decomposition of linear multi-degree-of-freedom systems with damping, circulatory, and potential forces is considered through a real linear coordinate transformation generated by an orthogonal matrix. Criteria are derived that establish the conditions under which such a transformation exists, allowing these systems to be decomposed into independent, uncoupled subsystems, each with a maximum dimension of two. These criteria are expressed in terms of the properties of systems' coefficient matrices. Several numerical examples are provided to demonstrate the analytical results.
. In this paper, we address the problem of the controlled motion of a roller racer on a plane. We assume that the angle between the platforms is a given periodic function of time (control function), and the no-slip conditions (nonholonomic constraint) and viscous friction forces act at the points of contact of the wheels with the plane. In this case, all trajectories of the reduced system tend asymptotically to a periodic solution. In this paper, we show that for a selected periodic control function, there exists a motion of the system that is bounded (along a circle) and unbounded (along a straight line). An unbounded motion corresponds to the resonant case which takes place at zero average value of the control function. The theoretical dependence of the trajectory and the velocity of the roller racer on its parameters and the parameters of the selected control function is investigated. These dependences are confirmed experimentally.
Dynamical systems more general than Hamiltonian systems are considered. The role of the Hamiltonian function is played by a 1-form (not necessarily closed) on a symplectic phase space. A bracket of such forms is introduced and a generalized Liouville theorem on the complete integrability is formulated. This generalization allows us to better understand the meaning of the conditions of the classical theorem on the complete integrability of the Hamilton equations and to reveal the role of tensor invariants.
Nipping analysis of a rectangular two-leaf spring strengthened by an additional full-length leaf is presented. Closed form expressions are derived for the initial gaps between the leaves which make the maximum stresses in all the leaves within the clamped cross-section of the loaded spring equal to each other. The derived expressions are general in the sense that they apply for any values of the introduced leaf-length and thickness parameters. Two initial gaps between the pairs of consecutive leaves are needed to achieve the desired stress reduction. The required gaps can be either positive or negative, depending on the values of introduced spring parameters. For some combination of length and thickness parameters, nipping is not an effective means of stress reduction. There is a particular combination of parameters for which the maximum stresses in all critical cross-sections of leaves become equal to each other. The presented analysis and obtained results may be useful for multi-leaf spring design and related optimization studies.
The problem of averaging on an infinite time interval is considered. The classical results on averaging proved by N.N. Bogoluybov are generalized to the case in which only a part of the coordinates in the phase space remains close to the equilibrium position of the averaged system. We call this the averaging with respect to a part of the coordinates. The results are based on some topological ideas combined with the standard theorem on averaging on a finite time interval.
